Recycling Probability and Dynamical Properties of Germinal Center Reactions

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Recycling probability and dynamical

properties of germinal center reactions

Michael Meyer-Hermann1, Andreas Deutsch2 and Michal Or-Guil2

1Institut fur Theoretische Physik, TU Dresden, D-01062 Dresden, GermanyE-Mail: meyer-hermann@physik.tu-dresden.de

2Max-Planck-Institut fur Physik komplexer Systeme,Nothnitzer Str.38, D-01187 Dresden, Germany

Abstract: We introduce a new model for the dynamics of centroblasts and centrocytes in agerminal center. The model reduces the germinal center reaction to the elements consideredas essential and embeds proliferation of centroblasts, point mutations of the correspondingantibody types represented in a shape space, differentiation to centrocytes, selection withrespect to initial antigens, differentiation of positively selected centrocytes to plasma ormemory cells and recycling of centrocytes to centroblasts. We use exclusively parameterswith a direct biological interpretation such that, once determined by experimental data,the model gains predictive power. Based on the experiment of Han et al. (1995b) we predictthat a high rate of recycling of centrocytes to centroblasts is necessary for the germinalcenter reaction to work reliably. Furthermore, we find a delayed start of the productionof plasma and memory cells with respect to the start of point mutations, which turns outto be necessary for the optimization process during the germinal center reaction. Thedependence of the germinal center reaction on the recycling probability is analyzed.

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1 Introduction

Germinal centers (GCs) are essential elements of the humoral immune response (for areview see MacLennan, 1994). In a primary immunization some B-cells respond to thepresented antigen and are activated. These active cells migrate to follicular dendriticcells (FDCs) present in lymphoid tissues. Unprocessed fragments of the antigen, so calledepitopes, are expressed on the FDCs. In the milieu of the FDCs the B-cells are in a phase ofintense proliferation and show the essential features of B-blasts such that they are denotedas centroblasts. These proliferating centroblasts together with the FDCs develop to a GC.

This morphological development is initiated through an unidentified differentiation sig-nal after about three days. The resulting GC is characterized by two zones (Nossal, 1991):the dark zone filled with continuously proliferating and mutating centroblasts and the lightzone containing the FDCs and centrocytes. The latter are generated from the centroblasts(Liu et al., 1991), which are available in great diversity (Weigert et al., 1970) because ofa very high mutation rate (Nossal, 1991; Berek & Milstein, 1987). In contrast to cen-troblasts the centrocytes express antibodies of cell specific types on their surface. In thisway they are able to undergo a specific selection process through the interaction with theepitopes presented on the FDCs in the light zone. The diversity of the proliferating andmutating centroblasts together with the specificity of the selection process in the light zoneallow an optimization of the affinity between the antibodies and the epitopes. This affinitymaturation in the GC leads to a dominant population of cells with high affinity of thecorresponding antibodies to the epitopes within 7-8 days after immunization (Jacob et al.,1993).

Centrocytes undergo apoptosis if they are not positively selected in an appropriaterange of time. They are rescued from apoptosis (Liu et al., 1989; Lindhout et al., 1993;Lindhout et al., 1995; Choe & Choi, 1998) if they are able to bind to the antigen on theFDCs and to interact successfully with T-helper cells present in the external region of thelight zone. Even antigen independent signals may be required for the centrocytes to survive(Fischer et al., 1998). Successfully selected centrocytes receive a differentiation signal whichdetermines their fate. They may differentiate into memory cells that are important in thecase of a second immunization. Alternatively, they differentiate to antibody producingplasma cells. These are intensifying the immune response in two ways: the number ofantibodies increases and their specificity with respect to the direct unspecific immuneresponse without GC reaction is optimized. In both cases the corresponding cell is leavingthe GC environment.

A third possibility exists for the positively selected centrocytes: they may be recycledto centroblasts and reenter the highly proliferating and mutating stage of developmentin the dark zone (Kepler & Perelson, 1993). This recycling hypothesis has been neitherdirectly checked experimentally (for an indirect check see Han et al., 1995b) nor – if true– the probability of recycling is known. It would be of great interest to know about thenecessity of a centrocyte recycling process for the affinity maturation. The most evidentargument to support the recycling hypothesis is based on the occurrence of multi-stepsomatic hypermutations in the GC reaction. Under the assumption of random hypermu-tations, recycling of already mutated cells to fast proliferating centroblasts appears to bea necessary process.

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In this paper we estimate how large the recycling probability should be, i.e. whichproportion of the positively selected centrocytes should be recycled to get a reliable GCreaction. For this purpose we introduce a simplified model of the GC dynamics based ona functional view on it. The model differs from others (Oprea & Perelson, 1996; Oprea &Perelson, 1997; Rundell et al., 1998) by omitting some details of the GC reaction which weconsider as unnecessary to understand the obligatory occurrence of the recycling procedure.On the other hand compared to the model introduced in (Kesmir & de Boer, 1999) weadded a formalism to treat somatic hypermutation in a shape space, which seems to beessential to understand the process of affinity maturation. A very important difference torelated models is that we embed the experiment (Han et al., 1995b) for the first time. Thisexperiment provides an indirect evidence for the existence of the recycling process and wewill interpret it in a quantitative way in order to extract information on the recyclingprobability from it. We use exclusively parameters with a direct biological interpretationto ensure the predictive power of the model.

In the next section we will define the shape space, in which the cell population dy-namics takes place, and the corresponding dynamical quantities and parameters includingtheir dynamical interdependence. All parameters introduced in the model are determinedby experimental observations. The third section is dedicated to the presentation and in-terpretation of the model in comparison with experimental GC reactions. We will analyzethe stability of our results to verify the notion of a prediction. Finally, we investigate thedependence of some characteristics of the germinal center reaction on the variation of therecycling probability and give some concluding remarks.

2 Definition of the model

2.1 Shape space and affinity space

Postulates Before defining a model for the GC dynamics it is necessary to specify thespace in which the dynamics takes place. As we want to represent cells corresponding todifferent types of antibodies we use the well known shape space concept (Perelson & Oster,1979). This ansatz is based on the following assumptions:

• Antibodies can be represented in a phenotypical shape space.

• Mutation can be represented in the shape space by next-neighbor jumps.

• Complementarity of antibodies and antigen.

• Homogeneity of the affinity weight function on the shape space.

Representation of antibody phenotype The shape space is taken as a D-dimensionalfinite size lattice of discrete equidistant points, each of them representing one specificantibody type. Thus the shape space becomes a phenotype space, i.e. it is not primarily arepresentation of genetic codes but of the resulting features due to specific genetic codes.

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Representation of mutation Nevertheless, we want to represent point mutations de-fined in an unknown genotype space in the shape space. To this end we are compelled toformulate the action of a point mutation in the shape space:A point mutation of a given antibody is represented in the phenotypical shape space by a

jump to one of the nearest neighbor points.

This assumption is not very spectacular as it only requires that a point mutation doesnot lead to a random jump of the antibody phenotype, i.e. that conformation, electricalproperties, etc. are not dramatically changed. Surely, the mutation of some key base pairsmay exist which imply fundamental changes of the encoded antibody features. However,these exceptions will not alter the dynamics of the GC until the number of such key basepairs remains small with respect to the number of smooth mutation points.

Antigen representation and complementarity A central quantity is the affinity of agiven antibody and an antigen epitope. Having a representation of antibodies in the shapespace, a counterpart for antigens is necessary. We emphasize that the number of possibleantibodies is finite, whereas the diversity of antigens is principally unbounded. Therefore,we represent an antigen on the B-cell shape space at the position of the B-cell with highestaffinity to the antigen, i.e. at the position of an antibody with optimal complementarity tothe antigen.

Property of smooth affinity Starting from these assumptions (phenotype shape space,representation of mutation by next-neighbor jumps, and complementarity) we have reacheda representation of antibodies and antigens in a shape space, which has the property ofaffinity neighborhood, i.e. that neighbor points in the shape space have comparable affinityto a given antigen. This property is a direct consequence of the definition of mutationsin a (phenotype) shape space. Starting from a non-directed mutation as base for affinitymaturation in GCs, we estimate the affinity neighborhood as a necessary property of theunderlying shape space. An initial B-cell with a particular affinity to a certain antigenmust have the possibility of successively optimizing the affinity, i.e. of stepwise climbingan affinity hill. If no affinity neighborhood would exist, i.e. if there was a random affinitydistribution on the shape space, each mutation would lead to a random jump in the affinity.An optimization of the affinity to the antigen may occur accidentally in this scenario, butshould be a very rare event.

Homogeneous affinity weight function Affinity neighborhood allows for the defini-tion of an affinity weight function which determines the affinity of antigen and antibody independence of their distance in the shape space. We assume this weight function a(φ, φ∗)to apply equally well in all regions of the shape space which corresponds to a homogeneousaffinity distribution over the shape space, and to be of exponential form:

a(φ, φ∗) = exp

(

−||φ − φ∗||η

Γη

)

(1)

where Γ is the width of the affinity weight function and || · || denotes the Euclidean metric.The question remains with which exponent η the distance enters the exponential weight

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function and it will be argued that η = 2 is a reasonable value (see Fig. 2).

2.2 Formulation of the dynamics

Having a shape space at hand it is possible to define distributions B(φ), C(φ), A(φ), andO(φ) of centroblasts, of centrocytes, of the presented antigen epitopes, and of the plasmaand memory cells on the shape space (see Tab. 1). We focus on the centroblast distributionB and analyze its dynamical behavior in the different phases of the GC reaction.

Cell type Variable Processesantigen A(φ) interaction with centrocytescentroblasts B(φ) proliferation

mutationdifferentiation to centrocytes

centrocytes C(φ) selection and apoptosisrecycling to centroblastsdifferentiation to plasma and memory cells

plasma and memory cells O(φ) removed from GC

Table 1: The variables in the mathematical model of the GC reaction and the processes thatare considered.

There were several attempts to divide the germinal center reaction in different workingphases. From the point of view of our model we are led to a new functional phase distinctionof the GC reaction (see Fig. 1). Our three-phase description of the GC reaction is inaccordance with most of the models established so far (see e.g.MacLennan, 1994; Liu etal., 1991; Kelsoe, 1996) but is in contradiction to the time scales found in (Camacho et al.,1998), which differ strongly. It is not intended to find mechanisms of transition betweenthe phases of the germinal center reaction. These dynamical phases are assumed accordingto experimental observations.

The first phase is the already mentioned phase of fast proliferation of a low numberof seeder cells in the environment of FDCs. In this phase it is likely that somatic hyper-mutation is not taking place to a relevant amount (Han et al., 1995b; Jacob et al., 1993;McHeyzer-Williams et al., 1993; Pascual et al., 1994b) such that it may be understood asan enlargement phase of the cell pool available for the following process of affinity matu-ration. The corresponding dynamical behavior of the centroblast distribution is describedas

dB

dt(φ) = pB(φ) for t − t0 < ∆t1 , (2)

where p denotes the proliferation rate and t0 is the time of first immunization. This phaselasts ∆t1 = 3 days and is well established by experiment (Liu et al., 1991).

After three days the optimization phase starts. The GC gets its morphological form,i.e. dark and light zones emerge. In the dark zone centroblasts continue to proliferate but,additionally, somatic hypermutations are broadening the initial centroblast distribution onthe shape space. At the same time the selection process operates in the light zone and

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Figure 1: The three model phases of the germinal center reaction: (I) proliferation phase(starting at t0 = −3 d), (II) optimization phase (starting at t1 = 0 d, the duration ∆t2 isdetermined in Sec. 2.8), (III) output phase (ending at t3 = 18 d). The parameters in bracketsrefer to the model equations Eq. (2), Eq. (3), Eq. (6), and Eq. (8) and can be determined fromexperimental data (see Sec. 2.2-2.8).

the selected centrocytes are either recycled to centroblasts or differentiate to plasma andmemory cells. The development of the centroblast distribution on the shape space is nowdescribed by

dB

dt(φ) = (p − 2pm − g)B(φ) +

pm

D

∑

||∆φ||=1

B(φ + ∆φ)

+(1 − q)∑

φ∗

C(φ)A(φ∗) a0 a(φ, φ∗) , (3)

for t − t0 ≥ ∆t1. Here, g is the rate of centroblast differentiation into centrocytes. m isthe mutation probability, a0 is the probability of an optimal centrocyte to be positivelyselected, q is the probability of differentiation into plasma and memory cells for positivelyselected centrocytes, and D is the dimension of the underlying shape space. The differentcontributions to Eq. (3) are discussed in detail below. Note that this equation does notinclude eventual finite size effects due to small populations.

It is not clear a priori, if differentiation into antibody producing plasma and memorycells is triggered already in this second phase. To allow a start of the output cell productiondelayed by the time interval ∆t2 we divide the optimization phase into two sub-phases

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which differ in the output probability q:

q = 0 for 0 ≤ t − ∆t1 < ∆t2

q > 0 for 0 ≤ t − ∆t2 < ∆t3 . (4)

The time delay ∆t2 will be fixed by experimental data (see Sec. 2.8). The output phase in-clude optimization and production of plasma and memory cells and lasts for the remainingGC life-time, which is about ∆t3 = 21days−3days−∆t2 (Choe & Choi, 1998; Kelsoe, 1996;Jacob et al., 1991). Then, only a few proliferating B-blasts remain in the environment ofthe FDCs (Liu et al., 1991).

Proliferation

During the whole GC reaction a fast proliferation of B-cells takes place. After the activationof B-cells by an interaction with an antigen, these move to FDCs and undergo a fastproliferation phase in their environment (Liu et al., 1991; Hanna, 1964; Zhang et al., 1988;Grouard et al., 1995). It is likely that the fast proliferation is triggered by the dendriticcells (Dubois et al., 1999). After about three days (Liu et al., 1991) when the GC starts todevelop its characteristic light and dark zone (Camacho et al., 1998), a fast proliferationof centroblasts is still observed in the dark zone. The extremely high rate of proliferationcould be determined to be (Liu et al., 1991; Zhang et al., 1988)

p

ln(2)≈

1

6 h, (5)

a result known already for a long time (Hanna, 1964). The population of centroblasts atφ grows exponentially, which is represented by the term pB(φ) in Eq. (2) and Eq. (3).

Mutation

It is likely that somatic hypermutation does not occur in the proliferation phase of the GCreaction (Han et al., 1995b; Jacob et al., 1993; McHeyzer-Williams et al., 1993; Pascualet al., 1994b). On the other hand the growth of the centroblast population is reduced bypossible mutations from state φ to its neighbors φ+∆φ, where ||∆φ|| = 1. In a continuousspace this corresponds to a diffusion process as used in (Perelson & Wiegel, 1999) torepresent mutation. Each pair of cells produced by proliferation will populate a neighborstate with a probability of m. This mutation probability turns out to take extremely highvalues of m ≈ 1/2 (Nossal, 1991; Berek & Milstein, 1987), which corresponds to a factor107 larger probability compared to mutations outside the GCs (Janeway & Travers, 1997).We want to point out that here only point mutations of phenotypical relevance are takeninto account. As a consequence the population growth by proliferation is reduced by theimportant amount 2pm, which results in an effective proliferation rate at φ of p(1 − 2m).One observes that for m = 1/2 the two first terms in Eq. (3) cancel. This corresponds tothe situation that in each centroblast replication one new cell remains in the old state andthe second new cell mutates to a neighbor state such that the total number of cells in stateφ remains unaltered.

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In the same way as mutation depopulates the state φ in the shape space and populatesits neighbors, φ is populated by its neighbors. This gives rise to the non-diagonal term inEq. (3). Each neighbor of φ mutates with the same rate 2pm from φ + ∆φ to one of its2D neighbors. Only the mutation from φ + ∆φ to the particular neighbor φ enhances thepopulation of centroblasts at φ such that the rate is weighted by the inverse number ofneighbors 1/(2D) and is summed over all possible neighbors of φ.

Selection, recycling and cell production

After the establishment of the light and dark zones in the GC the differentiation of cen-troblasts to antibody presenting centrocytes is triggered to allow a selection process in thelight zone. The centroblasts in the shape space state φ are diminished with the rate ofdifferentiation g, leading to the term −gB(φ) in Eq. (3). The centrocyte population C(φ)of type φ is enhanced simultaneously by the same amount of cells:

C(φ) = +gB(φ) . (6)

The centrocytes move to the light zone where they undergo a selection process. Theirfurther development is splitted three-fold. The non-selected centrocytes die through apop-tosis and are eliminated from the GC dynamics. It is known for a long time that thecentrocytes undergoing apoptosis were in cell cycle a few hours ago (Fliedner, 1967) suchthat it is likely that they differentiated from the centroblasts. On the other hand apoptosistakes place in the light zone (Hardie et al., 1993) where centroblasts are not present inhigh density. The selected centrocytes are emitted from the GC with probability q and dif-ferentiate either into antibody producing plasma cells or to memory cells. The model doesnot distinguish between plasma and memory cells. Only their sum is taken into accountand is denoted by output cells O(φ). Nevertheless, the dynamics of both types of outputcells may be different (Choe & Choi, 1998). Also, the degree of affinity maturation differs(Smith et al., 1997). In addition, we do not consider any further proliferation of outputcells in or outside of the GC, which may be important for a quantitative comparison withexperimental measurements.

Alternatively the selected centrocytes are recycled to become centroblasts and to reen-ter the proliferation process in the dark zone. This happens with probability 1 − q andcontributes to the centroblast distribution, corresponding to the last term in Eq. (3).

We want to emphasize that we do not resolve the time course of the selection process,which is regarded to be fast with respect to the centroblast proliferation time scale. Thisprocedure is justified by a minimal duration of a typical selection process of 1 − 2 hours(van Eijk & de Groot, 1999), which is about one fourth of the proliferation time scale.Nevertheless, one should be aware of possible implications due to the fact that the selectionprocess does not occur instantaneously. We effectively enclose the selection time in theparameter g, which in this way becomes a measure of a complete selection cycle includingthe differentiation into centrocytes, a first selection at the FDCs, a second selection atT-helper cells and finally the recycling process. This has two consequences for the model:The number of selectable centrocytes for each shape space state φ is given at every timeby Eq. (6), i.e. the number of centrocytes just being in the selection process. Furthermore,the details of the multi-step selection process (Lindhout et al., 1997) are omitted. The

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selection is modeled by a sum over the shape space of the antigen distribution presentedon the FDCs weighted by the affinity function Eq. (1). In other words the centrocytesgB(φ) are selected if an antigen is close enough in the shape space. The meaning of close

enough is determined by the width of the affinity function and by the probability a0 of apositive selection for an optimal centrocyte with respect to one presented antigen.

One should be aware that the probability a0 is not necessarily equal to one, as centro-cytes are in the state of activated apoptosis (Cohen et al., 1992) such that their life time isfinite and they have to be selected within this life time to be rescued from apoptosis. Thismaximum selection probability is determined by the relation of the centrocyte life time ofabout 10 h (Liu et al., 1989, 1994) and the duration of the selection process of 1− 2 h (vanEijk & de Groot, 1999). A rough estimate using two Gaussian distributions with reason-able widths for the life time and the selection time peaked at 10 h and 1.5 h respectivelyleads to a0 ≈ 0.95. Thus, if a GC contains only centrocytes of optimal complementarityto a presented antigen, still about 5% of them will undergo apoptosis.

In conclusion, selection can be described by the convolution term

∑

φ∗

C(φ)A(φ∗) a0 a(φ, φ∗) , (7)

where the number of selectable centrocytes is given by Eq. (6) and A(φ∗) represents the dis-tribution of antigens on the shape space. This selection term contributes with probabilityq to the production of plasma and memory cells

dO

dt(φ) = q

∑

φ∗

C(φ)A(φ∗) a0 a(φ, φ∗) . (8)

All selected centrocytes not emitted from the GC are recycled and in this way enhance thecentroblast population giving rise to the last term in Eq. (3).

2.3 Initial conditions

Antigen distribution

Throughout the paper the antigen distribution is assumed unequal to zero at exactly onepoint in the shape space:

A(φ∗) = A(φ0) = δ(φ∗ − φ0) , (9)

where δ(·) is one for its argument equal to zero and zero otherwise. Consequently, the sumin the convolution term in Eq. (3) is reduced to one contribution. This implies that weconsider only one type of antigen epitope to be present in the GC. What seems to be arestriction at first sight turns out to be an assumption without any loss of generality. Inview of about 1011 possible states in the naive repertoire of humans (Berek & Milstein,1988) two randomly chosen antigen epitopes will not be in direct neighborhood of eachothers in the shape space. Both antigen epitopes will only have an influence on each otherduring the GC reaction when centroblasts corresponding to one of the antigen epitopes havea non-negligible affinity to the other epitope. In the shape space language this situation

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corresponds to an overlap of two spheres centered at both antigen epitopes. The radius ofthese spheres is determined by the maximum number of mutations which may occur duringa GC reaction. This number will rarely exceed 9 (Kuppers et al., 1993; Wedemayer et al.,1997) such that in a 4-dimensional shape space (a more detailed discussion of the shapespace dimension follows in Sec. 2.5) one sphere of this size covers about 10−8th part ofthe space. An overlap is unlikely as an unreasonable large number of 108 antigen epitopesis needed to get an important probability for a mutual influence. Therefore we considerone-antigen epitope GC reactions only. A multiple-antigen or a multiple-antigen epitopeGC reaction has to be considered as sum of one-antigen epitope GCs.

There exists experimental evidence that the amount of presented antigens is reducedduring a GC reaction (Tew & Mandel, 1979). Since the amount of antigen is only halvedduring 30 days (Tew & Mandel, 1979; Oprea et al., 2000) we do not expect an importantrecoil effect on the selection probability, which would become time dependent otherwise.Furthermore, we believe that this antigen diminution is too weak to be responsible for thevanishing cell population in GCs after about 3-4 weeks. Anyhow, we aim to show thatour three-phase model allows for a vanishing population solely due to the dynamics of thesystem, without the inclusion of any antigen distribution dynamics.

B-cell distribution

The B-cells which have moved to FDCs are in a fast proliferation phase. In this first phaseof GC development the cell population grows exponentially until it has reached about 104

cells after three days. One may ask how many seeder cells are necessary and sufficientto give rise to this large cell population within such a short space of time. One findsexperimentally that the follicular response is of oligoclonal character (Liu et al., 1991;Jacob et al., 1991; Kuppers et al., 1993; Kroese et al., 1987) and the number of seeder cellsis estimated between two and six. This result is in accordance with the proliferation rateof ln(2)/(6 h) cited above.

The number of seeder cells∑

φ B(φ, t = 0) determines the initial B-cell distributionB(φ) in our model. Corresponding to the experimental results cited above, we start withthe reasonable number of three seeder cells for the follicular response to an immunizationwith a specific antigen. These cells are chosen randomly in the shape space within a sphereof radius R0 around the presented antigen. Due to the assumed affinity neighborhood thisreflects a threshold affinity of antigen and activated initial B-cells. This was already foundexperimentally (Agarwal et al., 1998): The activation of a B-cell by an antigen is necessarilyconnected with a minimum affinity of the corresponding epitopes and antibodies. Theradius of the sphere can be determined by the maximum number of mutations occurringduring the process of affinity maturation in GCs:

R0 = Nmax , (10)

which is found to be 8 or 9 (Kuppers et al., 1993; Wedemayer et al., 1997). In this picture,B-cells with an affinity above threshold are activated by a specific antigen, giving rise toa fast immune response with antibodies of low but non-vanishing affinity. The germinalcenter reaction has to be understood as an optimization process for these initially activatedB-cells leading to a second slower primary immune response of high specificity.

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2.4 Width of affinity function

Due to affinity neighborhood in the shape space the affinity function may be chosen ac-cording to Eq. (1). The width Γ has to be consistent with the affinity enhancement thatis achieved during a GC optimization process. Let the factor of affinity enhancement be10ξ and the number of corresponding somatic hypermutations be N . Then from Eq. (1)we get

Γ =N

(ξ ln(10))1

η

. (11)

Typical numbers of somatic hypermutations are N = 6 (Kuppers et al., 1993) correspond-ing to an affinity enhancement of 100 (ξ = 2) (Janeway & Travers, 1997), while N = 8is considered to be a large number of mutations (Kuppers et al., 1993) corresponding toa high factor of affinity enhancement of 2000 (ξ = 3.3) (Janeway & Travers, 1997). Ahuge affinity enhancement by a factor of 30000 (ξ = 4.5) is reached with N = 9 mutations(Wedemayer et al., 1997). As can be seen in Fig. 2, these values for the affinity enhance-ment with growing number of somatic hypermutations are consistent with a Gaussianaffinity function (η = 2) with width of Γ ≈ 2.8. It should be mentioned that a consistent

Figure 2: The width of the Gaussian affinity function as a function of the factor of affinity en-hancement for different numbers of somatic hypermutations. Vertical lines denote the positionsof the used experimental values of affinity enhancement. One can observe that consistently forall three values the resulting width becomes approximately 2.8.

value for the width of the affinity function can not be obtained for other integer values ofη. But one should be aware that the used affinity enhancements are very rough estimatesand thus our result should be understood as a first guess – even if the compatibility withthe experimental data is convincing.

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2.5 Shape space dimension

The shape space dimension D enters the model through the mutation term in Eq. (3)and has to be chosen appropriately to our model. We already mentioned that somatichypermutation is likely to occur randomly (Weigert et al., 1970; Radmacher et al., 1998),i.e. that the direction of mutation in the shape space is not governed by the position ofthe antigen. The term describing mutation in Eq. (3) is in analogy to a diffusion term ina continuous space as used in (Perelson & Wiegel, 1999). Let us assume for a momentthat the development of the GC reaction is governed by a diffusion process and let thenumber of maximum point mutations be Nmax. Then the diffusion process operates ina sphere around a seeder cell of radius Nmax in the shape space. In order to match theposition of the antigen with at least one centroblast, all positions in the sphere have to bereached by diffusion. This means that the total number of centroblasts Nc present in theGC 3 days after immunization, i.e. when selection is triggered, should be of the order ofmagnitude of the number of points in the sphere. The number of points in the sphere growsexponentially with the dimension of the shape space. Thus, for Nmax = 8 and Nc = 12000it follows that the shape space dimension should be 4. From the point of view of a purediffusion process this is an upper bound for the shape space dimension.

However, the above argument does not include the proliferation rate, the selection rate,and the recycling probability. The effective number of centroblasts available to reach theantigen may become larger than the 12000 cells assumed above through proliferation andrecycling. As we want to describe a decreasing population for large times without furtherassumptions, the effective rate of production of additional centroblasts during the GCreaction cannot exceed some small value, since the centroblast population would explodeotherwise for large times. So it seems unlikely that the shape space dimension appropriatefor our model is substantially larger than 4. We will start with D = 4 and check our resultsfor the dimensions 5 and 6. We want to remind that we are dealing with a phenotype spacein this model and that a genotype space probably requires a substantially higher dimension.

2.6 Long term behavior

The parameter g was introduced in the model as the rate of differentiation from centrob-lasts in the dark zone into antibody presenting centrocytes ready to undergo a selectionprocess. As we do not consider the centrocyte dynamics during the multi-step selectionprocess (Lindhout et al., 1997), this parameter effectively describes the total durationof the selection process including the differentiation mentioned above, the selection at theFDCs and with T-helper cells, and the recycling to centroblasts. Only those cells that havesuccessfully finished this program are contributing to the centroblast population describedin Eq. (3).

Thus, being an effective measure of the speed of the selection process, we are ableto deduce an upper bound for g. Solely the inhibition of apoptosis during the selectionprocess at FDCs and with T-helper cells takes at least 1-2 hours (Lindhout et al., 1995;van Eijk & de Groot, 1999). This gives us an upper bound for the rate g of

g

ln(2)<

1

2 h. (12)

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Nevertheless, this upper bound is not sufficient to fix g. As g governs the whole reactionspeed it will play a crucial role for the final state 21 days after immunization. As we knowthat at the end of the GC reaction only a few cells remain in the environment of the FDCs(Liu et al., 1991; Kelsoe, 1996), the parameter g will be tuned in order to get this result.

Mathematically, this requirement has the form of a long term condition. Let us lookfor an equilibrium condition at the point φ0 representing the antibody type of optimalcomplementarity to a presented antigen:

dB

dt(φ = φ0, t) = 0 . (13)

Then, the selection term in Eq. (3) becomes simply (1 − q)ga0B(φ0, t) and for a non-vanishing centroblast distribution at φ0 we get

ε ≡ p − 2pm − g +pmβ

D+ (1 − q)ga0 = 0 , (14)

where we have introduced ε as a measure for the vicinity to the equilibrium condition and

β(φ0, t) =

∑

||∆φ||=1 B(φ0 + ∆φ, t)

B(φ0, t). (15)

β(t) is a measure for the sharpness of the centroblast distribution in the shape spacearound the antigen position φ0. β becomes constant when the selection process startingat day 3 after immunization has reached a phase in which the centroblasts at the positionof the antigen dominate. This is already the case 8 days after immunization according toexperiments (Jacob et al., 1993) and is confirmed in our model (see Fig. 6).

Thus, at large times β can already be considered as constant in the equilibrium condi-tion Eq. (14) and the same applies to ε. In this stage of GC development the centroblastpopulation at φ0 has a pure exponential form

B(φ0, t) = eεt . (16)

The sign of ε determines whether the centroblasts population explodes (ε > 0) or dies out(ε < 0). Because of the finite life times of GCs, ε should adopt values slightly smaller thanzero.

2.7 Recycling probability

The recycling probability 1 − q determines the fate of the positively selected centrocytesin the light zone. Apoptosis was inhibited for these cells and they may either becomeplasma or memory cells or return to the dark zone to reenter the fast proliferation phase.This recycling hypothesis (Kepler & Perelson, 1993) has been intensively discussed andthe main position is that random somatic hypermutation – and it is likely that somatichypermutation occurs randomly (Weigert et al., 1970; Radmacher et al., 1998), i.e. thatthe direction of mutation in the shape space does not depend on the position of the antigen– does not lead to a sustained optimization of affinity in a one-pass GC reaction (Oprea

13

et al., 2000). To reach a specific position in the shape space a multi-step optimizationand verification process is required to avoid an arbitrary aimless walk through this highdimensional space.

To propose a new perspective onto this question we consider the most convincing ex-perimental evidence for the existence of such a recycling process (Han et al., 1995b). AGC reaction was initiated in mice with one antigen. 9 days after immunization, i.e. in astage of the reaction in which the good cells already dominate (Jacob et al., 1993), a seconddifferent, but related antigen was injected. This change of the reaction conditions in thecourse of the GC reaction leads to some very interesting observations: The rate of apop-tosis of centrocytes is enhanced by a factor of about ν ≈ 5 compared to the non-disturbedGC reaction. This can be explained by the fact that the antigen distribution in the shapespace was changed such that most of the cells present after 9 days may fit to the firstantigen but not to the second one. The probability of a positive selection is diminishedand the inhibition of apoptosis does not occur to the same extent as before. On the otherhand the total cell population in the GC vanishes with a ω ≈ 8-fold higher speed comparedto the unperturbed GC reaction. This is very difficult to explain without recycling: As weknow that the centrocytes (only very small numbers of centroblasts) undergo apoptosis ifnot rescued in time, the change of the apoptosis rate is of relevance only for the centrocytepopulation. The faster reduction of the whole GC cell population can not be explained bythe enhanced apoptosis rate, if there exists no recoil effect of the centrocyte apoptosis tothe centroblast population. This is an indirect confirmation of the recycling hypothesis.

We want to consider this argument in a quantitative way by translating this experimentinto the language of our model. We know from our consideration in Section 2.6 that at day 9after immunization the optimization process is already completed and that the centroblastpopulation at the antigen position φ = φ0 in the shape space behaves according to Eq. (16).But due to the modification of the antigen distribution at day 9 after immunization the GCreaction returns back to its dynamical selection phase. Thus, until day 9 the centroblastpopulation B1(φ, t) evolves with respect to the antigen distribution A1(φ

∗) as in Eq. (9) andafter day 9 two GC developments have to be compared: One continues its usual reactionas before. The second B2(φ, t) starts with the distribution B1(φ, t = 9 d) and evolves withrespect to the new antigen distribution

A2(φ∗) = ρ1δ(φ

∗ − φ0) + ρ2δ(φ∗ − (φ0 + ∆φ∗)) , (17)

where ∆φ∗ denotes the shift of the additional second antigen with respect to the first one.ρ1,2 take into consideration the (possibly) different concentrations of both antigens.

Now we are able to calculate the apoptosis rate in both scenarios for the cells of somefixed but arbitrary type φ. The number of centrocytes available for selection at everymoment is gB(φ). The apoptosis rate is given by this number reduced by the selected cells

Vi(φ) = gB(φ)

1 − a0

∑

φ∗

Ai(φ∗)a(φ − φ∗)

, (18)

where i denotes the two scenarios. For the original GC reaction this becomes

V1(φ) = gB(φ) (1 − a0a(φ − φ0)) , (19)

14

while in the GC with the new antigen distribution we find

V2(φ) = gB(φ) (1 − a0a(φ − φ0)ρ1 (1 + α(φ))) , (20)

where we have shifted φ∗ in the second term of Eq. (17) and introduced the ratio

α(φ) =ρ2a(φ − φ0 − ∆φ∗)

ρ1a(φ − φ0)(21)

valid for each point φ separately. The factor ν of the apoptosis enhancement found in theexperiment above is then given by the ratio of both apoptosis rates

ν =V2

V1

=1 − a0a(φ − φ0)ρ1 (1 + α(φ))

1 − a0a(φ − φ0). (22)

On the other hand, this enhancement of apoptosis leads to a faster population reductionby a factor ω, which in our model is given by the ratio of the changes of the cell distributionsover the shape space at the time of the presentation of the new antigen. According toEq. (3) and using the new antigen distribution A2 this ratio is given by

ω =dB2

dt

[

dB1

dt

]−1

(φ, t = 9 d)

=p − 2pm − g + pm

Dβ(φ) + (1 − q)ga0a(φ − φ0) ρ1 (1 + α(φ))

p − 2pm − g + pm

Dβ(φ) + (1 − q)ga0a(φ − φ0)

, (23)

where the cell distributions B(φ, t = 9 d) cancel as they are equal in both scenarios fort = 9 d.

Taking the results for ν (Eq. (22)) and for ω (Eq. (23)) at the shape space point of theprimary antigen φ = φ0 the affinity function becomes equal to one. β is already a constantfor t = 9 d such that we may eliminate ρ1(1 + α(φ0)) from both equations to get

ν − 1

ω − 1=

p − 2pm − g + pm

Dβ(φ0) + (1 − q)ga0

(1 − q)g(a0 − 1)(24)

or for the recycling probability

1 − q = −p − 2pm − g + pm

Dβ(φ0)

g[

a0 + (1 − a0)ν−1

ω−1

] . (25)

This relation allows us to determine the parameter q and in this way to give a quantitativeprediction for the recycling probability for selected centrocytes to reenter the dark zoneand to continue proliferation. The recycling probability has to fulfill Eq. (25) in order tobe consistent with the experiment of Han et al., (1995b). We want to emphasize, that thecondition Eq. (25) is independent of the concentrations of the two antigen types used inthe experiment.

15

2.8 Output dynamics

The production of plasma and memory cells optimized with respect to the presented antigenin the course of the GC reaction is governed by Eq. (8). According to our discussion of thedifferent phases of the GC reaction in Sec. 2.2, the starting time of output cell productionis not necessarily correlated with the starting time of the mutation and selection processabout 3 days after immunization. There may occur a time delay of the output production inrelation to the selection process (see Fig. 1). This possibility is encountered by comparingthe dynamics of output cell production with experiment (Han et al., 1995a). Here, thenumber of output cells of high affinity to the presented antigen φ0 is compared at day 6and day 12 after immunization and their relation is found to be approximately

vO ≡

∫

12d0d dt O(φ0)∫

6d0d dt O(φ0)

≈ 6 . (26)

As output cells are already present at day 6 after immunization, the starting point of outputcell production has to fulfill 0 h < ∆t2 < 72 h (∆t2 denotes the time window between thestart of mutation and selection and the start of output cell production, see Fig. 1). It isintuitively clear that for large ∆t2 the speed of output cell production vO will be enhanced,for the cell population of the GC will already be peaked at the position of the antigen inthe shape space. Thus, we are able to adjust ∆t2 to the correct output cell dynamics.

3 Results and Predictions

Parameter quantity = valueProliferation rate p/ ln(2) = 1/(6 h)Somatic hypermutation probability m = 0.5Rate of differentiation of centroblasts to centrocytes g/ ln(2) = 0.352Dimension of shape space D = 4Selection probability for optimal centrocytes a0 = 0.95Output probability of selected centrocytes q = 0.2Number of seeder cells at FDCs

∑

φ B(φ, t = 0) = 3Radius of B-cell activation around antigen R0 = 8 − 10Time duration of phase I of GC reaction ∆t1 = 72 hTime duration of phase II of GC reaction ∆t2 = 48hTime duration of the whole GC reaction

∑

3

i=1 ∆ti = 504 h

Table 2: Summary of all parameters of the model. They were determined by experimentaldata. For explanations and references see the last section. The parameters g, q, and ∆t2 aredetermined in the course of the solution of Eq. (3) with respect to experimental data discussedbefore.

Starting from a randomly chosen antigen epitope and – following the oligoclonal char-acter of the B-cell population of a GC – three randomly chosen activated B-cells in a sphere

16

Figure 3: Time course of the centroblast population integrated over the whole sphere in theshape space around the antigen φ0. In the first phase the initial B-cells proliferate. The largepopulation reached at t = 0 h is then reduced by the selection process. At t = 48 h theproduction of plasma and memory cells starts, leading to an exponential decrease in the overallcentroblast population.

of radius R0 according to Eq. (10), we let the GC reaction develop as described by Eq. (2)and Eq. (3). The different phases (see Sec. 2.2) of the GC reaction are respected. Theparameters are determined from experimental data as summarized in Tab. 2. The param-eters g, 1 − q and ∆t2, describing the differentiation rate of centroblasts to centrocytes,the recycling probability for the selected centrocytes and the time delay for the productionof output cells, respectively, are iteratively adjusted in order to fit with the experimentsdescribed in the previous section. The time variable is running from t0 = −72 h to accen-tuate the first phase of pure proliferation as a preGC-phase. The mutation and selectionprocess is started at t = 0 h and the production of plasma and memory cells begins att = ∆t2. The differential equation is solved numerically with a modified Euler method ina subspace, in which we assume Dirichlet boundary conditions with B = 0.

The best results are obtained by an iteration procedure for

g

ln(2)=

0.355

h, 1 − q = 0.8 , ∆t2 = 48 h . (27)

At t = 0 h we find a large oligoclonal population of 12288 centroblasts as a result of theproliferation phase started with 3 initial cells (see Fig. 3). These cells are of different typecompared to the cells of optimal affinity to the presented antigen and in a typical distance

17

from them (varying between 3 and 8 point mutations to reach the optimal cell type).

Optimization phase The selection process at the FDCs and with the T-helper cellsstarts and reduces the centroblast population with low affinity to the presented antigen.At the same time somatic hypermutation leads to a spreading of the centroblast distributionover the shape space, resulting in a small but non-vanishing amount of centroblasts at theshape-space position of the antigen (see Fig. 4). The large majority of B-cells does not

Figure 4: The centroblast population and the integrated plasma/memory cell production at theposition of the antigen in the shape space. The centroblast population shows a growth due toproliferation and somatic hypermutation starting at t = 0 h. The population growth is sloweddown by the production of output cells starting at t = 48 h. Now the number of producedplasma and memory cells is increased steadily until the end of the GC reaction. Then, only 2cells (of optimal type) remain in the environment of the FDCs.

survive the selection process (compare the summed cell population decrease in Fig. 3), diesthrough apoptosis and is rapidly phagocytosed by macrophages present in the GC.

Secondary proliferation As all positively selected centrocytes reenter the proliferationprocess in the optimization phase of the GC reaction, the number of centroblasts at theantigen position grows. Note that ε > 0 follows for the equilibrium condition Eq. (14) be-cause of q = 0, according to Eq. (4). As a consequence, the overall centroblast populationwould restart to grow if this growth was not inhibited by the production of plasma andmemory cells after 48 h. Especially, this secondary proliferation process is found for the

18

centroblasts encoding antibodies of high affinity to the antigen (see Fig. 4). The centrob-lasts at the antigen position are practically all recycled (beside the reduction due to a0 < 1)such that no reduction process exists any more. This gives rise to a new perspective onthe GC reaction: We need just as many initially proliferated cells, that a spreading of thedistribution through somatic hypermutation leads to at least one or two cells matching theantigen position in the shape space. Then a considerable amount of cells of this optimizedtype is reached within the time delay ∆t2 of the production of output cells, opening thepossibility of a full proliferation process for these cells.

Recycling probability When the output production is turned on at t = 48 h the dy-namics of the germinal center is characterized by ε < 0 in Eq. (14), such that an exponentialdecrease of the whole centroblast population is initiated – including the population of theoptimized cells (see Fig. 3 and Fig. 4). Nevertheless, this decrease is slow enough to allowthe production of a considerable amount of plasma and memory cells (see Fig. 4 and Fig. 5).It is an interesting observation that a rather small number of centroblasts of the optimal

Figure 5: A cut through the sphere in the shape space around the presented antigen at position2. The seeder cells are at a distance of at least 5 mutation steps. The number of producedplasma and memory cells grows considerably during the GC reaction. The rapidity of growthdiminishes in course of time because of the decreasing number of optimized centroblasts at theantigen position.

type with respect to the antigen is sufficient to produce this large number of plasma andmemory cells – even with a small output probability of q = 0.2. On the contrary, a larger

19

output probability may decrease the integrated number of plasma and memory cells, asless cells reenter the proliferation process. In conclusion, we claim that a large recyclingprobability of 80% of the positively selected centrocytes is not only necessary to achievean optimization by multi-step mutations but also to achieve a high number of resultingplasma and memory cells.

The end of the GC reaction Finally, at day 21 after immunization, only a few cellswith maximum affinity to the antigen remain in the environment of the FDCs, as requiredfrom experiment (Liu et al., 1991; Kelsoe, 1996). So the dynamics of our model allowfor a vanishing of the germinal center population without any reduction of the antigenconcentration or other additional requirements. The length of the germinal center reactionis determined by the interplay of the centroblast (into centrocyte) differentiation rate gand the delay of output production ∆t2, which is determined by Eq. (26), so that the GClife time is basically controlled by g. So from the perspective of our model a diminish-ing cell population may result solely from the interplay of proliferation, differentiation ofcentroblasts to centrocytes, and output production.

Dependence of initially activated seeder cells The total number of remaining cellsdepends on the distance of the initial B-cells from the antigen φ0. For numbers of mutationsteps (necessary to reach the antigen) between N = 3 and N = 8, the number of remainingB-blasts varies from 43 to 0. For a typical number of mutations of N = 5 about 10 B-blasts remain in the environment of the FDCs at day 21 after immunization. In otherwords, the GC life time depends on the initial distribution of the seeder cells with respectto the presented antigen. It would be interesting to check this statement experimentallyby observing GCs with a variable number of mutations occurring during the optimizationprocess. One could imagine for real GCs a self-regulating process which depends on thealready positively selected centrocytes and thus prolongs the GC reactions with largedistances between seeder cells and antigen, and shortens correspondingly the ones withsmall distances. However, there is experimental evidence that the intensity of the GCreaction depends on the quality of the initially activated B-cells (Agarwal et al., 1998).Also in quasi-monoclonal mice (Cascalho et al., 1996) it was found that the volume of GCs4 days after immunization depends on the average affinity of the reservoir B-cells to theantigen (de Vinuesa et al., 2000, Fig. 6). A quantitative comparison of these data withour model results would be interesting.

3.1 Result stability

The parameter set in Eq. (27) is determined such that the experimental conditions Eq. (25)and Eq. (26) are fulfilled for typical initial conditions. Therefore, the prediction of therecycling probability to be 80% is a statement which – in the framework of our model – isjustified to the same degree as the experiments are exact. Nevertheless, the sensitivity ofthe predicted high recycling probability to such modifications is weak.

It should be mentioned that in addition there is a certain freedom of choice concerningthe number of remaining cells at day 21 after immunization. For example, the parameter

20

set with a still higher recycling probability

g

ln(2)=

0.52

h, 1 − q = 0.9 , ∆t2 = 55 h . (28)

leads to a result of comparable quality as under the conditions of Eq. (27) – it is clearthat a higher recycling probability slows down the production of output cells such thatsimultaneously the time delay of the production start has to be larger – with the differencethat the number of remaining cells is reduced with respect to the previous result such thatno cells remain at the end of typical GC reactions. For this reason and because g is on itsupper bound Eq. (12), this parameter set was not considered as reasonable.

If one requires a smaller recycling probability the parameter set

g

ln(2)=

0.286

h, 1 − q = 0.7 , ∆t2 = 42 h . (29)

leads to correct values for v0 and Eq. (25) is also fulfilled. The reduced recycling probabilityaccelerates the output cell production such that it has to be decelerated by a reduction ofthe time delay ∆t2. Even if the differentiation rate stays in a reasonable range, the numberof remaining cells in the GC at day 21 after immunization is about ten-fold with respectto Eq. (27). More generally one may consider the number of remaining cells for typical GCreactions with an average number of mutations necessary to reach the antigen.

Recycling probability 1 − q 0.5 0.6 0.7 0.8 0.9∑

φ B(φ, t = 21 d) 1476 515 134 10 0

The mentioned experimental data are all reproduced for each value of 1 − q, solely thenumber of remaining cells is left open. To ensure that this number is in accordance withthe experimental statement that a a few proliferating B-blasts remain at the FDCs atthis stage of GC development, we are led to a recycling probability of 80%. We want toemphasize that this result is determined very clearly, as for the other recycling probabilitiesthe remaining number of cells in the GC final state differs by at least an order of magnitude.

This variation of parameters shows the range of stability of our prediction:First, the window of possible time delays ∆t2, which are in accordance with Eq. (26) is verynarrow. Already for ∆t2 > 55 h the condition Eq. (12) will be violated. On the other handfor ∆t2 < 38 h, despite the immense number of remaining cells in the final GC state, thecondition Eq. (26) can not be fulfilled anymore. In consequence, it is not possible that theproduction of plasma and memory cells already starts with the establishment of light anddark zone, i.e. that it starts with the existence of a fully developed GC. The emphasizedexperimental bounds require necessarily a time delay for the production of output cells ofat least 42 h. This means that the production process should be initiated by a separatesignal and is not present in a working GC automatically.Secondly, the window of possible recycling probabilities is strongly bound by the experi-ments (Han et al., 1995a, 1995b) and by the experimentally observed number of remainingcells at day 21 after immunization. This leads to the conclusion that the recycling proba-bility should be at least 70% and not larger than 85%. Even ignoring the required numberof cells in the final state, Eq. (26) does not allow a recycling probability smaller than 60%.

21

We would like to emphasize that this result differs extremely from the value for the recy-cling probability of 0.15 assumed in (Kesmir & de Boer, 1999). Nevertheless, we do notregard this discrepancy as a contradiction because in the model of Kesmir & de Boer notall parameters were fixed by experimental data.

Shape space dimension The analysis presented above was performed with a shapespace dimension of 4, (compare Sec. 2.5). It is an important remark that our results remainessentially unaffected for D = 5 or 6. Using for instance D = 5, only the parameterg, controlling the selection speed, has to be reduced slightly by 7% in order to stay inaccordance with the experimental bounds. The total number of remaining cells at day 21after immunization does not change but its distribution is spread out on the shape space.For D = 6, the parameter g is again reduced by 6%, while all other parameters remainunchanged to stay in accordance with the experimental data. Again, the cell distributionat day 21 after immunization becomes less centered at the antigen position. We concludethat, for the values tested, the dependence on the shape space dimension is weak enoughsuch that our results are not affected by it. Even, the slower selection speed for higherdimensions of less than 10% per dimension remains in the range of accuracy of our results.

Note, that the dependence on the shape space dimension is restricted to an intermediatestage (phase II) of the GC reaction. In the first phase of pure proliferation D does not enterthe dynamics at all. In the late phase of selection, when the distribution B(φ) becomesapproximately spherically symmetric around the point representing the antigen, D cancelsexactly with the factor 1/D in Eq. (3) as there are D equal terms contributing to the sumsuch that D factorizes. A dependence of our results on the shape space dimension mayonly occur during the early phase of selection, in which the distribution B(φ) is clearlynot symmetric around the position of the antigen. Since the distribution of centroblastsdevelops into a spherically symmetric distribution around φ0 already before day 8 afterimmunization (see Fig. 6), we deduce that a dependence on the shape space dimension mayoccur between day 3 and day 8 of the GC reaction only.

Mutations with large changes of affinity We have assumed so far that mutationsare represented in the shape space by next-neighbor jumps. To check the robustness of theresult for the recycling probability against the possibility of mutations that lead to largedistance jumps in the shape space, we consider a worst case argument. Let us suppose that10 % of the mutations lead to a large jump in the shape space. This is translated into themodel by reducing the number of next-neighbor-mutations by a factor of 0.9. The resultis a lower efficiency of the mutation process as the probability for a random mutation toimprove affinity with respect to the antigen is very small. Therefore, the final number ofcells is reduced by a factor of 10, and the centroblast to centrocyte differentiation rate hasto be adjusted (from 0.244 to 0.255) in order to fulfill condition Eq. (25). A similar resultas before is obtained by an adjustment of the recycling probability from 0.8 to 0.78. As thisis a worst case argument, we consider our result as robust against large jump mutations.

22

Figure 6: The time course of β(φ0, t) (see Eq. (15)) at the shape space position of the antigenfor the parameter set Eq. (27). β becomes constant shortly after t = 96 h, i.e. at day 7 afterimmunization.

3.2 Optimization speed

From experiment it is expected that the cells with high affinity to the antigen dominatealready at day 8 after immunization (Jacob et al., 1993). This is verified in the model bychecking if the asymptotic regime of the GC reaction is reached at day 8 in the sense thatβ (defined in Eq. (15)) becomes constant. As can be seen in Fig. 6, β becomes constant inthe course of day 7 after immunization for the parameter set Eq. (27). This behavior ofβ is not altered in a great range of parameter variation. Besides the reproduction of theexperimental evidence for the optimization process to be accomplished after 8 days, thisresult is an a-posteriori check for the derivation of the recycling probability, which is basedon a constant value of β(φ0) at day 9 after immunization (see Eq. (24)).

3.3 Optimization quality

It may be interesting to verify if larger or smaller recycling probabilities lead to betteroptimization of the antibody types. The statement that recycling may be necessary toachieve a considerable affinity enhancement in a multi-step mutation process is widelydiscussed (Oprea et al., 2000) and enforced by the present work. We would like to point outthat the relative number of good outcoming plasma or memory cells increases with largerrecycling probabilities (see Fig. 7). In the shape space this corresponds to a sharper peak

23

Figure 7: Comparison of the total plasma and memory cell distribution at day 21 after immu-nization on a cut along one of the coordinates in the shape space with the antigen at coordinateposition 2. The parameters are uniquely determined by the experimental data with the excep-tion of the remaining number of cells at the end of the GC reaction, which is undeterminedto allow different recycling probabilities 1 − q. The distribution of the output cells becomessharper for larger 1 − q.

of the distribution∫

21 d0 d dt O(φ, t). A too small recycling probability leads to a spreading

of the outcoming plasma and memory cells in the shape space, i.e. to a weaker specificityof the GC output.

On the other hand, we have already seen that a larger recycling probability lowers theabsolute number of produced plasma and memory cells such that we are confronted withtwo competitive tendencies. Large recycling probabilities lead to specific but weak GCreactions, while small probabilities lead to unspecific but intense GC reactions. One mayagree, that the value of 80% calculated here and supported by experimental data, is a goodcompromise between these two extremes.

3.4 Start of somatic hypermutation

There is experimental evidence that somatic hypermutation does not occur during the firstphase of proliferation of centroblasts in the environment of the FDCs (Han et al., 1995b;Jacob et al., 1993; McHeyzer-Williams et al., 1993; Pascual et al., 1994b). However, wechecked if our model allow somatic hypermutation to start during the proliferation phase.The primary reason for the retardation of the output production is to give time to the GC

24

to develop sufficient good centroblasts before the GC population is weakened through anadditional output. Therefore, one may in principle think of a mutation starting long beforethe selection process. In this scenario the output production would start simultaneouslywith selection and one is led to a two phase process only: A first one with proliferation andsomatic hypermutation and a second one with additional selection and output production.The centroblast-types would already be spread out in the shape space when selectionand output start. There is no parameter set in accordance with all experimental data.

Figure 8: The centroblast population and the integrated plasma/memory cell production atthe position of the antigen in the shape space. Except for the time phases we used the sameparameters as in Eq. (27). The centroblast population shows a smooth enhancement due tosomatic hypermutation starting at t = −48 h. The population growth is stopped by theproduction of output cells starting at t = 0 h. The slope of the summed output is substantiallysmaller than expected from experiment. 21 days after immunization only 1 cell remains.

Especially, it is impossible to get the correct output dynamics as described in Eq. (26).Typically we get v0 = 2.9 being strongly different from the required value and a muchsmoother result (compare Fig. 8 to Fig. 4). In addition the distribution of plasma andmemory cells is not as well pronounced at the antigen position such that the GC reactiongives rise to output cells with weaker specificity. Thus, we deduce from our model thatthe dynamic of plasma and memory cell production in real GC reactions does not allowsomatic hypermutation to occur considerably before the selection process is started.

25

4 Conclusions

We developed a new model for the GC reaction using its main functional elements. TheGC reaction is described by the evolution of a centroblast distribution on an affinity shapespace with respect to an initially fixed antigen distribution. On this shape space an affinityfunction, modeling the complementarity of antibody and antigen, was defined and itsfunctional behavior as well as its width were deviated from affinity enhancements knownfor real GC reactions.

In the model the reaction is decomposed into three phases (see Fig. 1). In the pro-liferation phase a few seeder cells multiply. Our model excludes somatic hypermutationto occur already at this stage of the GC development. An optimization phase follows, inwhich in addition to proliferation, mutation of the antibody type and selection take place.This leads to a competition of spreading and peaking of the centroblast distribution inthe shape space. In this phase, all positively selected cells reenter the proliferation phasein the dark zone, i.e. no plasma or memory cells are produced. This occurs solely in theoutput production phase. In this third phase all elements of the GC reaction are active.

The dynamical evolution of the cell population in a GC is described by a set of coupledlinear differential equation. All parameters are determined with experimental data andespecially using (Han et al., 1995b). The model shows the typical behavior of GCs withoutany further adjustments. Due to the oligoclonal character of GCs a large amount ofidentical centroblasts is produced in the proliferation phase. These diffuse over the shapespace in the second phase, in which only those cells are retained that exhibit a highaffinity to the presented antigen. As the good cells reenter the proliferation process, thisleads to a dominance of good cells within 7 to 8 days after immunization in accordancewith experiments. The already optimized cell distribution then gives rise to plasma andmemory cells leaving the FDC environment in the output production phase, leading tolarge numbers of output cells of optimized type due to the long duration of this thirdphase. The GC reaction is rather weak 21 days after immunization and only a smallnumber of cells remains in the environment of the FDCs. This shows that the end of a GCreaction is not necessarily coupled with an antigen consumption or additional signals, butmay occur simply due to the (unchanged) dynamical development of the GC.

Our main result is that the experimental data and especially the evaluation of (Hanet al., 1995b) lead to the following view of the GC reaction. The optimization phase lastsfor not less than 42 h and not longer than 55 h. During this time all selected centrocytesare recycled and reenter the proliferation phase. Without such a non-output phase thenumber of optimal cells with respect to the presented antigen is not large enough to allowa considerable output rate during the remaining life time of the GC. We looked for anexperimental evidence for this delayed production of output cells. Indeed, comparingtwo experiments (Jacob et al., 1993; Pascual et al., 1994a) we found that plasma andmemory cells were first observed more than two days later than mutated cells during a GCreaction. This confirms experimentally the existence of the non-output phase that we foundin the model. As output cells are not produced from the beginning of the GC reaction,the output should be triggered by a special signal, related to the affinity of the B-cellsand the corresponding antigen. But even in the output production phase, the recyclingprobability turns out to lie between 70% and 85% and thus is substantially larger than

26

values expected until now (Kesmir & de Boer, 1999). We want to emphasize that suchlarge recycling probabilities turned out to be crucial for a GC outcome of high specificity.Larger recycling probabilities lead to specific GC reactions of low intensity, while smallerones to less specific but more intense GC reactions. The value of about 80% that we foundrepresents a good compromise between these competitive tendencies. On the other hand itis not surprising that these recycled cells have not been found experimentally: The absolutenumber of recycled cells is very small compared to the total number of cells present in theGC, so that a measured signal for recycled cells is suppressed by more than two orders ofmagnitudes. A better signal should be observed in later stages of the GC reaction whenthe cells of high affinity to the antigen are already dominating.

Furthermore, we conclude that the whole selection process including differentiation ofcentroblasts to antibody presenting centrocytes, the multi-step selection process itself andfinally the recycling take around 3 or 4 h. This result is compatible with the experimentalobservation that the inhibition of apoptosis during the selection may take about 1 or 2 h.

It is interesting, that the produced cells possess a certain broadness in the shape space,i.e. not only the centrocytes of optimal affinity to the antigen differentiate to plasma andmemory cells. This may be of relevance for the resistance of an immune system againsta second immunization with a mutated antigen. A considerable number of memory cellswith high affinity to the mutated antigen may still be present for this reason.

The results of our model suggest new experiments. A systematic and quantitativeanalysis of the GC reaction in dependence of the initial conditions, i.e. the number ofnecessary point mutations to reach the cell type of optimal affinity to the antigen, wouldlead to new insights into the interdependence of the essential elements of the GC. This mayyield hints about self-regulating processes. Furthermore, it would be interesting to analyzequantitatively the total number of cells in the course of the GC reaction. Finally, it maybe interesting to compare our model with its continuous counterpart. The translation ofEq. (3) into continuous space leads to a differential equation of Schrodinger type with aGaussian potential.

27

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